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Full text of "Treatise On Analysis Vol-Ii"

440         XV    NORMED ALGEBRAS AND SPECTRAL THEORY

8.    A conjugation in a complex Hilbert space E is a semilinear bijection C of E onto E
(i.e.,  such  that   C-(ax +fty) = aC-x +&C-y)   satisfying   (C-x\C-y) = (y\x)
and C~l = C. Show that if a closed operator TonE commutes with C (which implies
that C(dom(r)) <= dom(r)), and if dom(T) is dense in E, then C also commutes with
7**. Deduce that, if a hermitian operator H commutes with C, then C(E&) = EH, and
consequently the defects of H are equal. Apply this result in particular to the case
where E is obtained by extension of scalars from a real Hilbert space E0, and H
is the extension to E of an unbounded operator H0 defined on a dense subspace
dom(#o) of E0ğ such that (H0 • x \ y) = (x \ H0 - y) for all *, y in dom(#o).

9.    Show that a surjective hermitian operator H is self-adjoint. (For each y e dom(#*),
remark that there exists z e dom(H) such that H* • y = H - z, and show that z must
equal y.)

10.   Let E be an infinite-dimensional separable Hilbert space.

(a)    If (an) is any infinite sequence in E, then the vector subspace generated by the
an (i.e. the set of all (finite) linear combinations of the an) is not equal to E (cf. (12.16.1)).

(b)    Show that there exist two Hilbert bases (an)t (bn) of E such that, if F and G
are the vector subspaces of E generated by the an and the bn respectively, we have
F n G == {0}. (Starting with an arbitrary Hilbert basis (Ğ„), construct inductively
a total sequence (cn) such that the subspace G it generates is such that F n G = {0},
by using (a). Then orthonormalize the sequence (cn).)

(c)    Suppose that the Hilbert bases (<?„), (/>„) have the property stated in (b).
Consider two compact self-adjoint operators A, 3 on E, defined by A - an — Xnan
and B -bn = Xnbn, where (Aw) is a sequence of real numbers >0 and tending to 0.
Show that the sequence (An) can be chosen so that A(E) n B(E) = {0}. (Proceed by

induction: let Sn be the set of points of F of the form j] £*A*<2fc with V |£

and let dn be the distance (strictly positive) of Sn from the subspace Gra generated by
blt..., bn; choose the A* with k > n such that ]T Afc < dn/n. Then prove that, if U is

k>n

the closed ball with center 0 and radius 1 in E, we have A(U) r\ B(U) = {0}.)

(d)    The operators A and B are injective; A ~1 and B~l are therefore self-adjoint closed
operators (Problem 8) such that dom^"1) n domO??"1) = {0},

(e)    Deduce from (d) an example of a self-adjoint operator H and a unitary operator
U on E such that U2=1E and dom(H) r^ dom(U'1HU) = {Q}. (Take E = FİF
and tf - (x, y) = (yt x), and take H to be equal to A'1 on one of the summands F,
and to B"1 on the other, where A and B are defined as above.) Hence give an example
of a closed operator Tsuch that dom(T') is dense in E, but dom(r2) = {0}.

11. Let Tbe a closed operator on a separable Hilbert space E. The essential spectrum of
Tis the set of £ e C such that Im(r— £/) is not closed; it is a subset of the spectrum
of T.

(a)    Show that if TV is an unbounded normal operator on E, then the isolated points
of Sp(TV) do not belong to the essential spectrum of TV. (If A is isolated in Sp(TV) and
if M = Sp(TV) — {A}, show that Im(TV — A/) is the image of E under the projector
P~ <pM(AO (Section 15.12, Problem 7), by observing that there exists a continuous
bounded function / on C such that /(TV) • y e dom(TV) and (TV — A/) • (/(TV) -y) — y
forau>eP(E).

(b)    Conversely, show that for each unbounded self-adjoint operator A on E, a point
A e Sp(A) which does not belong to the essential spectrum of A is isolated in Sp(A) to a measure which does not belong to M£(X, v).